Continued Fraction Identities

Theorem

First Continued Fraction Identity

• Let $\left[{a_1, a_2, a_3, \ldots, a_n}\right]$ be the continued fraction expansion of a simple finite continued fraction.

Then

$\displaystyle \left[{a_1, a_2, a_3, \ldots, a_n}\right] = a_1 + \frac 1 {\left[{a_2, a_3, \ldots, a_n}\right]}$.

• Let $\left[{a_1, a_2, a_3, \ldots}\right]$ be the continued fraction expansion of a simple infinite continued fraction.

Then

$\displaystyle \left[{a_1, a_2, a_3, \ldots}\right] = a_1 + \frac 1 {\left[{a_2, a_3, \ldots}\right]}$.

Second Continued Fraction Identity

• Let $\left[{a_1, a_2, a_3, \ldots, a_n, a_{n+1}}\right]$ be the continued fraction expansion of a simple finite continued fraction.

Then

$\displaystyle \left[{a_1, a_2, a_3, \ldots, a_n, a_{n+1}}\right] = \left[{a_1, a_2, a_3, \ldots, a_n + \frac 1 {a_{n+1}}}\right]$.

Note that $\displaystyle \left[{a_1, a_2, a_3, \ldots, a_n + \frac 1 {a_{n+1}}}\right]$ is not in general a simple continued fraction because $\displaystyle \frac 1 {a_{n+1}}$ is not necessarily an integer.

Proof

These both follow directly from the definition.